Introduction
Two-Way Tables for Categorical Data is an important Grade 8 math skill because students are moving from simple answers toward explaining how the math works.
In this lesson, students use models, real questions, worked examples, practice problems, and two online quizzes to build confidence with two-way tables for categorical data.
What Is Two-Way Tables for Categorical Data?
Two-Way Tables for Categorical Data means reading, creating, and explaining displays so data can answer real questions.
The goal is not only to get the answer. Students should be able to show the idea, explain the strategy, and check whether the answer makes sense.
Understanding Two-Way Tables for Categorical Data
Before solving, students should slow down and decide what each number, shape, unit, or label represents.
- Read the title, labels, and scale before answering.
- Use the scale value instead of counting marks as ones when the graph is scaled.
- Compare categories by subtracting or adding values from the display.
- Explain what the data shows in a complete sentence.
Visual Models
Visual Model 1
Question: A two-way table shows that \(30\) students like pizza and \(20\) like sandwiches. \(15\) of those who like pizza are girls. What is the relative frequency of girls among pizza-likers?
| Girls | Total | |
|---|---|---|
| Pizza | 15 | 30 |
| Sandwiches | -- | 20 |
- A. \(50\%\)
- B. \(60\%\)
- C. \(30\%\)
- D. \(75\%\)
Why it works: This is a row-conditional relative frequency. Among pizza-likers (30 total), 15 are girls: \(\frac{15}{30}=0.5=50\%\).
Answer: \(50\%\)
Visual Model 2
Question: The table below shows preferences for sports among 100 students. How many boys prefer basketball?
| Football | Basketball | Total | |
|---|---|---|---|
| Boys | 35 | 20 | 55 |
| Girls | 15 | 30 | 45 |
| Total | 50 | 50 | 100 |
- A. 15
- B. 20
- C. 35
- D. 50
Why it works: This is a joint frequency (read directly from the cell). Row label Boys, column Basketball: entry is 20.
Answer: 20
Worked Examples
Example 1
Question: A two-way table of \(120\) people shows: What is the joint frequency of rural people who own a car?
| Owns Car | No Car | Total | |
|---|---|---|---|
| Urban | 24 | 36 | 60 |
| Rural | 45 | 15 | 60 |
| Total | 69 | 51 | 120 |
- A. \(\frac{60}{120}\)
- B. \(\frac{45}{60}\)
- C. \(\frac{45}{120}\)
- D. \(\frac{45}{69}\)
- Joint frequency: intersection cell (45) divided by grand total (120).
- This answers 'what fraction of all respondents are rural car-owners?' = \(\frac{45}{120}\approx0.375\).
Answer: \(\frac{45}{120}\)
Example 2
Question: A two-way table categorizes 150 households by income level and internet access: Among high-income households, what is the relative frequency of those without internet, as a decimal?
| Has Internet | No Internet | Total | |
|---|---|---|---|
| Low Income | 40 | 50 | 90 |
| High Income | 55 | 5 | 60 |
| Total | 95 | 55 | 150 |
- A. \(0.033\)
- B. \(0.083\)
- C. \(0.367\)
- D. \(0.917\)
- Among high-income households (total 60), those without internet = 5.
- Relative frequency = \(\frac{5}{60}\approx0.083\).
Answer: \(0.083\)
Example 3
Question: A two-way table shows 160 athletes: What percentage of all athletes train in the morning on the track team?
| Morning | Evening | Total | |
|---|---|---|---|
| Track | 50 | 30 | 80 |
| Swim | 35 | 45 | 80 |
| Total | 85 | 75 | 160 |
- A. \(31.25\%\)
- B. \(37.5\%\)
- C. \(58.8\%\)
- D. \(62.5\%\)
- Joint frequency of morning track athletes: \(\frac{50}{160}=0.3125=31.25\%\).
Answer: \(31.25\%\)
Real-World Word Problems
Problem 1
Question: Of 200 high school students surveyed, \(120\) play sports and \(80\) do not. Among those who play sports, \(72\) are boys. What fraction of sports players are girls?
- A. \(\frac{48}{200}\)
- B. \(\frac{48}{120}\)
- C. \(\frac{72}{120}\)
- D. \(\frac{48}{80}\)
Why it works: This is a row-conditional frequency. Girls in sports \(=120-72=48\). Relative to sports players only: \(\frac{48}{120}=\frac{2}{5}\).
Answer: \(\frac{48}{120}\)
Problem 2
Question: A school newsletter reports that 250 students were surveyed about lunch preferences. The two-way table shows: What is the marginal frequency for middle school students?
| Packed Lunch | School Lunch | Total | |
|---|---|---|---|
| Elementary | 60 | 65 | 125 |
| Middle | 55 | 70 | 125 |
| Total | 115 | 135 | 250 |
- A. 55
- B. 70
- C. 115
- D. 125
Why it works: The marginal frequency for middle school is the row total: \(55+70=125\).
Answer: 125
Common Mistakes
- Ignoring the graph scale.
- Reading the wrong category or axis label.
- Answering a comparison question without subtracting.
- Writing a number without explaining what it represents.
Strategy Tips
- Circle the scale before using the graph.
- Write down the value for each category you compare.
- Use addition for totals and subtraction for differences.
- Answer in words so the data result has meaning.
Practice Questions
Question 1
In a survey of 80 coffee drinkers, 50 add cream and 30 do not. Of those who add cream, 35 are office workers. What is the relative frequency of office workers among cream-adders, rounded to the nearest whole percent?
- A. \(44\%\)
- B. \(70\%\)
- C. \(62\%\)
- D. \(56\%\)
Question 2
In a survey of 250 voters, 120 voted yes on a proposition and 130 voted no. If 75 of the yes-voters are women, what is the relative frequency of men among yes-voters as a percent?
- A. \(37.5\%\)
- B. \(30\%\)
- C. \(62.5\%\)
- D. \(48\%\)
Question 3
In a study of 240 employees, 144 are full-time and 96 are part-time. Among full-time employees, 108 have health insurance. What is the relative frequency of uninsured full-time employees?
- A. \(\frac{36}{240}\)
- B. \(\frac{36}{144}\)
- C. \(\frac{108}{240}\)
- D. \(\frac{108}{144}\)
Question 4
A restaurant tracks 300 orders by meal type and customer type. Of 180 dine-in orders, 108 are from regular customers. What is the relative frequency of orders from regular customers among dine-in meals, as a decimal?
- A. \(0.36\)
- B. \(0.45\)
- C. \(0.54\)
- D. \(0.60\)
Question 5
In a survey of 200 people on movie preferences, 120 prefer comedy. Of those who prefer comedy, 84 are female. What is the relative frequency of females among comedy-lovers?
- A. \(0.42\)
- B. \(0.60\)
- C. \(0.70\)
- D. \(0.84\)
Question 6
Among 300 shoppers surveyed, 180 shop online and 120 shop in-store. Of those who shop online, 108 use mobile apps. What percentage of online shoppers use mobile apps?
- A. \(36\%\)
- B. \(48\%\)
- C. \(60\%\)
- D. \(72\%\)
Full Answer Explanations Click to show all answers and explanations
Question 1
Answer: \(70\%\)
This is a row-conditional relative frequency. Among cream-adders (50 total), 35 are office workers: \(\frac{35}{50}=0.70=70\%\).
Question 2
Answer: \(37.5\%\)
Men voting yes \(=120-75=45\). Relative frequency \(=\frac{45}{120}=0.375=37.5\%\).
Question 3
Answer: \(\frac{36}{144}\)
Uninsured full-time \(=144-108=36\). Relative frequency among full-time only: \(\frac{36}{144}=\frac{1}{4}=0.25\). This is row-conditional.
Question 4
Answer: \(0.60\)
Relative frequency among dine-in orders: \(\frac{108}{180}=0.60\). This is a row-conditional frequency.
Question 5
Answer: \(0.70\)
Relative frequency of females among comedy-lovers: \(\frac{84}{120}=0.70\). This is a row-conditional frequency.
Question 6
Answer: \(60\%\)
Relative frequency of mobile-app users among online shoppers: \(\frac{108}{180}=0.60=60\%\).
Connection to Standards
This lesson supports Grade 8 math expectations for reasoning, modeling, problem solving, and explaining answers clearly. It connects classroom skills to the kind of questions students see on state math assessments.
Summary
Two-Way Tables for Categorical Data becomes easier when students connect the question to a model, use clear steps, and explain why the answer fits.
GOLDEN RULE
Read the scale before reading the answer.
